GHM method for obtaining rationalsolutions of nonlinear differential equations

Vázquez-Leal, H.; Sarmiento-Reyes, A.

doi:10.1186/s40064-015-1011-x

Cited by 4 publications

(4 citation statements)

References 44 publications

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“…It exploits the well-known concept of multiparameter homotopy continuation methods to the homotopy perturbation techniques. (v) e generalized homotopy method [16,17] is a powerful generalization of the homotopy perturbation method [18][19][20]. It consists on replacing the traditional power series (v 0 + v 1 p + v 2 p 2 + • • •) of the homotopy parameter p by a power series of a trial function with respect to p. is paradigm change converts HPM into an especial case of GHM.…”

Section: Introductionmentioning

confidence: 99%

Exploring the Novel Continuum-Cancellation Leal-Method for the Approximate Solution of Nonlinear Differential Equations

Vázquez-Leal

2020

Discrete Dynamics in Nature and Society

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This work presents the novel continuum-cancellation Leal-method (CCLM) for the approximation of nonlinear differential equations. CCLM obtains accurate approximate analytical solutions resorting to a process that involves the continuum cancellation (CC) of the residual error of multiple selected points; such CC process occurs during the successive derivatives of the differential equation resulting in an accuracy increase of the inner region of the CC-points and, thus, extends the domain of convergence and accuracy. Users of CCLM can propose their own trial functions to construct the approximation as long as they are continuous in the CC-points, that is, it can be polynomials, exponentials, and rational polynomials, among others. In addition, we show how the process to obtain the approximations is straightforward and simple to achieve and capable to generate compact, and easy, computable expressions. A convergence control is proposed with the aim to establish a solid scheme to obtain optimal CCLM approximations. Furthermore, we present the application of CCLM in several examples: Thomas–Fermi singular equation for the neutral atom, magnetohydrodynamic flow of blood in a porous channel singular boundary-valued problem, and a system of initial condition differential equations to model the dynamics of cocaine consumption in Spain. We present a computational convergence study for the proposed approximations resulting in a tendency of the RMS error to zero as the approximation order increases for all case studies. In addition, a computation time analysis (using Fortran) for the proposed approximations presents average times from 3.5 nanoseconds to 7 nanoseconds for all the case studies. Thence, CCML approximations can be used for intensive computing simulations.

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Section: Introductionmentioning

confidence: 99%

Exploring the Novel Continuum-Cancellation Leal-Method for the Approximate Solution of Nonlinear Differential Equations

Vázquez-Leal

2020

Discrete Dynamics in Nature and Society

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“…Therefore, the obtained numerical solution may not represent the desired result [3,4]. Hence, a line of research is focused to obtain solutions in the form of analytical approximations like: enhanced power series method [5], power series extender method [6,7], modified Taylor series method [8], direct Padé method [9,10], rational homotopy perturbation method [11], generalized homotopy method [12,13], homotopy analysis method [2,14], homotopy perturbation method [15,16,17], among others. On one hand, all above mentioned methods may provide approximate solutions in terms of polynomials; nevertheless, users require specialized knowledge of mathematics to apply such methods.…”

Section: Introductionmentioning

confidence: 99%

The novel Leal-polynomials for the multi-expansive approximation of nonlinear differential equations

Vázquez-Leal

Sandoval-Hernandez

Filobello-Nino

et al. 2020

Heliyon

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This work presents the novel Leal-polynomials (LP) for the approximation of nonlinear differential equations of different kind. The main characteristic of LPs is that they satisfy multiple expansion points and its derivatives as a mechanism to replicate behaviour of the nonlinear problem, giving more accuracy within the region of interest. Therefore, the main contribution of this work is that LP satisfies the successive derivatives in some specific points, resulting more accurate polynomials than Taylor expansion does for the same degree of their respective polynomials. Such characteristic makes of LPs a handy and powerful tool to approximate different kind of differential equations including: singular problems, initial condition and boundary-valued problems, equations with discontinuities, coupled differential equations, high-order equations, among others. Additionally, we show how the process to obtain the polynomials is straightforward and simple to implement; generating a compact, and easy to compute, expression. Even more, we present the process to approximate Gelfand's equation, an equation of an isothermal reaction, a model for chronic myelogenous leukemia, Thomas-Fermi equation, and a high order nonlinear differential equations with discontinuities getting, as result, accurate, fast and compact approximate solutions. In addition, we present the computational convergence and error studies for LPs resulting convergent polynomials and error tendency to zero as the order of LPs increases for all study cases. Finally, a study of CPU time shows that LPs require a few nano-seconds to be evaluated, which makes them suitable for intensive computing applications.

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“…Several authors applied HPM to solve linear and nonlinear partial differential equations of fractional order (Momani and Odibat 2007a , b ), Volterra integral equations (Grover and Tomer 2011 ), singularly perturbed Volterra integral equations (Alnasr and Momani 2008 ) and n-th order fuzzy linear differential equations (Tapaswini and Chakraverty 2013 ). The reader is also referred to two recent papers where the authors (Filobello-Nino et al 2014a ; Vazquez-Leal and Sarmiento-Reyes 2015 ) make use of HPM in their application oriented works. In the very recent studies, it has been found that homotopy perturbation method was also used to solve fractional fisher’s equation (Hamdi Cherif et al 2016 ) and a system of nonlinear chemistry problems (Ramesh Rao 2016 ).…”

Section: Introductionmentioning

confidence: 99%

Homotopy perturbation method: a versatile tool to evaluate linear and nonlinear fuzzy Volterra integral equations of the second kind

Narayanamoorthy

Sathiyapriya

2016

SpringerPlus

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In this article, we focus on linear and nonlinear fuzzy Volterra integral equations of the second kind and we propose a numerical scheme using homotopy perturbation method (HPM) to obtain fuzzy approximate solutions to them. To facilitate the benefits of this proposal, an algorithmic form of the HPM is also designed to handle the same. In order to illustrate the potentiality of the approach, two test problems are offered and the obtained numerical results are compared with the existing exact solutions and are depicted in terms of plots to reveal its precision and reliability.

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GHM method for obtaining rationalsolutions of nonlinear differential equations

Cited by 4 publications

References 44 publications

Exploring the Novel Continuum-Cancellation Leal-Method for the Approximate Solution of Nonlinear Differential Equations

Exploring the Novel Continuum-Cancellation Leal-Method for the Approximate Solution of Nonlinear Differential Equations

The novel Leal-polynomials for the multi-expansive approximation of nonlinear differential equations

Homotopy perturbation method: a versatile tool to evaluate linear and nonlinear fuzzy Volterra integral equations of the second kind

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